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I've been going through "Online Interactive Collaborative Filtering Using Multi-Armed Bandit with Dependent Arms" by Wang et al. and am unable to understand how the update equations for the hyperparameters (section 4.3, equation set (23)) were derived. I'd deeply appreciate it if anyone could provide a full or partial derivation of the updates. Any general suggestions regarding how to proceed with the derivation would also be appreciated.

ICTR Graphical Model

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The variables are sampled as below

$$\mathbb{p}_m|\lambda \sim \text{Dirichlet}(\lambda)$$

$$\sigma^2_n|\alpha,\beta \sim \text{Inverse-Gamma}(\alpha,\beta)$$

$$\mathbb{q}_n |\mu_{\mathbb{q}}, \Sigma_{\mathbb{q}}, \sigma_n^2 \sim \mathcal{N}(\mu_{\mathbb{q}}, \sigma_n^2\Sigma_{\mathbb{q}})$$

$$\mathbb{\Phi}_k |\eta \sim \text{Dirichlet}(\eta)$$

$$z_{m,t} | \mathbb{p}_m \sim \text{Multinomial}(\mathbb{p}_m)$$

$$x_{m,t} | \mathbb{\Phi}_k \sim \text{Multinomial}(\mathbb{\Phi}_k) $$

$$y_{m,t} \sim \mathcal{N}(\mathbb{p}_m^T\mathbb{q}_n, \sigma_n^2)$$

And the update equations are below

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